J-T AND J-Q CHARACTERIZATION OF SURFACE CRACK TIP FIELDS IN METALLIC LINERS UNDER LARGE-SCALE YIELDING

Transcription

1 J-T AND J-Q CHARACTERIZATION OF SURFACE CRACK TIP FIELDS IN METALLIC LINERS UNDER LARGE-SCALE YIELDING By SHAWN A. ENGLISH A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA

2 2011 Shawn A. English 2

3 Whatever you do, work at it with all your heart, as working for the Lord, not for men. Colossians 3:23 3

4 ACKNOWLEDGMENTS The author expresses his gratitude to Dr. Nagaraj Arakere for his guidance, support and friendship as the graduate advisor on this project. Thanks also to the graduate committee members: Dr. John Mecholsky, Dr. Peter Ifju, Dr. Ashok Kumar and Dr. Ghatu Subhash for reviewing the work. Special thanks goes to Dr. Phillip Allen and others at the NASA Marshall Space Flight Center (MSFC). Ongoing, unpublished research at MSFC laid the foundation for many of the analytical tools and techniques used in this study. A very special thanks goes to the author s wife, Lisa, for her selflessness and unwavering support. The author also thanks his fellow colleagues: Dr. Nathan Branch and Dr. George Levesque for their challenging discussions and willingness to review the work. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS... 4 LIST OF TABLES... 8 LIST OF FIGURES... 9 LIST OF ABBREVIATIONS NOMENCLATURE ABSTRACT CHAPTER 1 INTRODUCTION Composite Overwrapped Pressure Vessels NDE, Proof Test Logic and Smart Cracks Coupon and Subscale Testing Two-Parameter Fracture Mechanics J-T and J-Q Characterization of Near Tip Fields Fracture Prediction Deformation Limits BACKGROUND Linear Elastic Fracture Mechanics Elastic-Plastic Fracture Mechanics Two-Parameter Characterization J-T Theory J-Q Theory Modified Boundary Layer Formulation Near Tip Stress Field Analysis and Plastic Collapse FINITE ELEMENT MODELING AND TEST GEOMETRIES Geometries Finite Element Models Mesh Configuration Mesh Convergence Element Formulation

6 4 DEFORMATION LIMITS AND PLASTIC COLLAPSE IN A BONDED GEOMETRY Model Definitions Materials Finite Element Models J-T Stress Field Characterization and Deformation Limits J-Integral and T-Stress Opening Stress Fields Deformation Limits Concluding Remarks STRAIN HARDENING EFFECTS ON TWO-PARAMETER CHARACTERIZATION Model Definitions Materials Finite Element Models Results and Discussion J-Integral and Q Crack Tip Deformation Near Tip Opening Stress Fields Concluding Remarks SURFACE-CRACKED METALLIC LINERS IN COMPOSITE OVERWRAPPED PRESSURE VESSELS Models Definitions Materials Geometries and Loads Results and Discussion J-Q and Critical Crack Angle Near Tip Field Characterization Limits Constraint Loss and Triaxiality in the Full-Scale COPV Concluding Remarks FRACTURE PREDICTION WITH TWO-PARAMETERS Model Definitions and Experimental Methods Material Experimental Methods J-integral and Constraint Factors Two-Parameter Stress Field Characterization Concluding Remarks SUMMARY

7 APPENDIX A POST-PROCESSING PSEUDO-CODE B MATERIAL PROPERTY INPUTS Ramberg-Osgood Linear Plus Power-Law T6 Tensile Data C LINEAR PLUS POWER-LAW DERIVATION D INTERPOLATION METHODS E COMPOSITE MATERIAL PROPERTIES LIST OF REFERENCES BIOGRAPHICAL SKETCH

8 LIST OF TABLES Table page 4-1 Dimensions of models used in the analysis (a = mm (0.035 in)) Surface crack model dimensions Dimensions of the metallic liner models Dimensions of the COPV models J-Q dominance limit loads at ϕ = Points of large scale yielding (J/J EL = 1.10) Q at ζ = ζ o Crack dimensions, FEA measured J and Q values and critical angles J Max and Q Max values and angles (ζ MBL (T = 0, r = 2J/ζ o ) = 2.75) B-1 Initial and final stress and plastic strain inputs for a LPPL model B-2 Stress and plastic strain inputs 6061-T6 aluminum

9 LIST OF FIGURES Figure page 1-1 Examples of common COPV geometries Stress-strain curve for the auto-frettage cycle of a COPV Picture of a COPV used for fracture testing Surface crack face from a subscale COPV liner t = 2.29 mm (0.09 in.) Outer strain gage measurements from COPV test specimen with FEA comparisons Diagram showing the relationship between constraint and deformation for various fracture characterizations Plastic zone shapes diagram for high (T/ζ o = 1), zero (T/ζ o = 0) and low (T/ζ o = -1) T-stresses Modified boundary layer (MBL) model mesh and displacement boundary conditions MBL generated near tip opening stresses as a function of normalized radial distance r/(j/ζ o ) for a series of T-stresses ranging from T/ζ o = 0.4 to Relationship between normalized T-stress and Q Reference opening stress as a function of T/ζ o (A) and Q (B) for normalized radial distances r/(j/ζ o ) = 2, 4, 6, and Examples of the near tip opening stress fields at a constant far field stress ζ = ζ o as a function of normalized radial distance r/(j/ζ o ) General uniaxial surface crack model configuration Example of crack front collapsed face nodal configuration Finite element surface cracked mesh refinement detail Data obtained from a single edge notched tensile (SEN(T)) specimen demonstrating the effects of using linear (small-strain) and non-linear (largestrain) kinematic element formulations Ramberg-Osgood material models for a low (n = 40) strain hardening material and 6061-T6 aluminum test data General bonded surface crack model configuration

10 4-3 J-integral as a function angle ϕ at a constant nominal load ζ = ζ o T-stress scaling factor as a function of angle ϕ Opening stress normalized by MBL reference solutions as a function of normalized T-stress (T/ζ o ) at r/(j/ζ o ) = 2 at ϕ = 90 (A) and 30 (B) Opening stresses normalized by MBL reference solutions as a function of normalized nominal stress (ζ ) at r/(j/ζ o ) = 2 at ϕ = 90 (A) and 30 (B) Opening stresses normalized by MBL reference solutions as a function of deformation factor aζ o /J at r/(j/ζ o ) = 2 at ϕ = 90 (A) and 30 (B) Opening stresses normalized by MBL reference solutions as a function of angle ϕ at r/(j/ζ o ) = 2 and ζ /ζ o = 0.95 (A) and ζ /ζ o = 1.0 (B) Normalized nominal load ζ /ζ o at opening stresses corresponding to plastic collapse (ζ yy /ζ MBL = 0.95) at various crack angles ϕ Normalized J-integral (J/(aζ o )) at opening stresses corresponding to plastic collapse (ζ yy /ζ MBL = 0.95) at various crack angles ϕ Linear plus power law material models for the high (n = 3) and low (n = 20) strain hardening materials used in the surface crack analysis General surface crack model configuration J-integrals as a function of crack angle ϕ at a constant average net section stress ζ Net = ζ o Q as a function of crack angle ϕ at a constant average net section stress ζ Net = ζ o for a/t = 0.70 and 0.35 and aspect ratios a/c = 1.0 (A) and 0.40 (B) Local deformation factor aζ o /J at crack angle ϕ = 30 as a function of normalized average net section stress ζ Net /ζ o Schematic representation of the relationship between the crack tip blunting displacement (u y ) and CTOD in a typical crack tip from this analysis Crack tip blunting displacement u y verses J-integral at crack angle ϕ = 30 for a/t = 0.70 and 0.35 for aspect ratios a/c = 1.0 (A) and 0.40 (B) Opening stresses normalized by the MBL predicted stresses at a constant normalized radial distance r/(j/ζ o ) = 4 and crack angle ϕ = 30 as a function of normalized average net section stress ζ Net /ζ o Opening stresses normalized by the MBL predicted stresses at a constant normalized radial distance r/(j/ζ o ) = 4 and crack angle ϕ = 30 as a function of local deformation factor aζ o /J

11 5-10 Opening stresses at an angle ϕ = 30 as a function of normalized radial distance r/(j/ζ o ) for a load aζ o /J = 40 (large deformation) Opening stresses normalized by the MBL predicted stresses at a constant normalized radial distance r/(j/ζ o ) = 4 as a function of crack angle ϕ Uniaxial stress strain data for 6061-T6 aluminum liner material and Abaqus material inputs for incremental plasticity Surface crack from sub-scale COPV General COPV liner section surface crack model configuration Strain data from the sub-scale COPV specimen and 3D FEA results FEA visualization of the surface cracked liner sub-model J (A) and Q (B) crack front distributions for the sub-scale COPV and liner models at a nominal equivalent stress ζ = ζ o J (A) and Q (B) crack front distributions for the full-scale COPV and liner models at a nominal equivalent stress ζ = ζ o The product of J(ϕ) and opening stress at a radial distance (r = 2J/ζ o ) normalized by the maximum J normalized by J IC as a function of nominal equivalent liner stress (ζ ) for angles ϕ = 24 (A) and 90 (B) in the sub-scale COPV and liner models J normalized by J IC as a function of nominal equivalent liner stress (ζ ) for angles ϕ = 24 (A) and 90 (B) in the full-scale COPV and liner models Opening stress (ζ yy ) normalized by J-Q MBL predicted stress at r = 4J/ζ o as a function of nominal equivalent liner stress ζ (A) and local deformation factor aζ o /J (B) at ϕ = 24 in the sub-scale models Opening stress (ζ yy ) normalized by J-Q MBL predicted stress at r = 4J/ζ o as a function of nominal equivalent liner stress ζ (A) and local deformation factor aζ o /J (B) at ϕ = 90 in the sub-scale models Opening stress (ζ yy ) normalized by J-Q MBL predicted stress at r = 4J/ζ o as a function of nominal equivalent liner stress ζ (A) and local deformation factor aζ o /J (B) at ϕ = 24 in the full-scale models Opening stress (ζ yy ) normalized by J-Q MBL predicted stress at r = 4J/ζ o as a function of nominal equivalent liner stress ζ (A) and local deformation factor aζ o /J (B) at ϕ = 90 in the full-scale models T-stress scaling factors (T/ζ h )

12 6-16 Q and Q(T) as a function of local deformation aζ o /J Q and Q(T) as a function nominal equivalent stress ζ The ratio of mean stress to von-mises stress at a radial distance r = 2J/ζ o verses Q for the full liner and COPV models J, ζ Q and Jζ Q crack front distributions normalized by their respective maximums Opening stresses at the experimental critical crack angle (ϕ Crit ) as a function of normalized radial distance r/(j/ζ o ) C-1 Schematic of the transition region in a linear plus power-law material model C-2 Material constant K 2 uniquely defined as a function of strain hardening exponent n in the linear plus power-law model D-1 Example of cubic spline interpolation of nodal points for a surface crack geometry and a linear plus power-law material E-1 Example of angle-ply orthotropic elastic stiffnesses as a function of transformation angle θ E-2 Example of angle-ply orthotropic Poisson s ratios as a function of transformation angle θ E-3 Example of angle-ply orthotropic shear moduli as a function of transformation angle θ

13 LIST OF ABBREVIATIONS ASTM CT COPV CMOD CTOD EDM FAD LARC LSY LEFM MEOP MBL NESC NASA NDE SENB SENT SSY SCB SCT American Society for Testing and Materials Compact Tension Composite Overwrapped Pressure Vessel Crack Mouth Opening Displacement Crack Tip Opening Displacement Electrical Discharge Machining Failure Assessment Diagram Langley Research Center Large-Scale Yielding Linear Elastic Fracture Mechanics Maximum Expected Operating Pressure Modified Boundary Layer NASA Engineering & Safety Center National Aeronautics and Space Administration Non-Destructive Evaluation Single Edge Notched Bend Single Edge Notched Tensile Small-Scale Yielding Surface Cracked Bend Surface Cracked Tensile 13

14 NOMENCLATURE t L t b t Hel t Hoop L w a c r d i d o β J T Q thickness of the metallic section thickness of the bonded section for uniaxial specimens thickness of the helical overwrap layer thickness of the hoop overwrap layer half length half width crack depth half crack width radial distance from crack tip inner pressure vessel diameter outer pressure vessel diameter wrap angle measured from the longitudinal direction on the outer pressure vessel surface J-integral T-stress near crack stress difference constraint parameter ζ Q ζ y normalized opening stress (ζ yy /ζ o ) in the crack plane at r = 2J/ζ o 0.2% offset yield strength ζ o reference yield stress (commonly ζ o = ζ y ) ε o n ζ ζ Net yield strain strain hardening exponent nominal stress of metallic section (Uniaxial: opening direction, Biaxial: Von-Mises) average stress in the opening direction over the remaining area in the crack plane 14

15 δ u y θ ϕ far-field displacement crack tip blunting displacement angle in plane perpendicular to crack front, θ = 0 is in the growth direction angle measured along the crack front in the crack plane 15

16 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy J-T AND J-Q CHARACTERIZATION OF SURFACE CRACK TIP FIELDS IN METALLIC LINERS UNDER LARGE-SCALE YIELDING Chair: Nagaraj K. Arakere Major: Mechanical Engineering By Shawn A. English May 2011 Composite overwrapped pressure vessels (COPV) are widely used in aerospace applications. Understanding surface crack behavior in COPV metallic liners is challenging because, (i) the liner experiences initial plastic deformation during manufacturing processes, (ii) the liner may experience cyclic plastic strains during normal operation, and (iii) surface crack behavior may be strongly influenced by the constraint effects of the bonding. Methodology is developed to assess the constraint effects imposed by the backing on the surface crack tip stress fields in the liner. Near tip stress fields in uniaxial tensile loaded metallic liner specimens are developed using J-T characterization and modified boundary layer (MBL) solutions, where J measures the asymptotic field and T measures constraint. The increased elastic constraint imposed by the backing on the liner results in enhanced validity of J-T characterization. In addition, presented is a rigorous study of strain hardening effects on near tip stress fields in surface cracks using J-Q MBL reference fields and near tip deformation estimates, where Q is a stress difference constraint parameter. At moderate loads, the radial independence of Q cannot be assured for a low strain hardening material. In 16

17 ASTM geometric standard development, this effect must be considered before a conservative limit is applied. These methodologies are applied to sub-scale and full-scale COPV models. J-Q fracture prediction of surface cracks in COPV metallic liners is found to be possible up to large deformations. Surface cracks in bonded geometries produce higher near tip triaxiality with respect to constraint level for large deformations. This indicates that while J-Q and J-T predicted stress fields are valid at large loads in these regions, the nonlinearity may hinder Q as a ductile fracture parameter. The results from this study aim to address the feasibility of using a two-parameter fracture theory to predict failure and characterize limits of both surface crack specimens and COPV liners. These results and methodologies are of practical value for establishing testing standards for surface crack geometries, development of proof test logic for COPVs and broader establishing of methodologies for assessing near tip dominance, parameterization deformation limits and plastic material property effects in fracture prediction. 17

18 CHAPTER 1 INTRODUCTION Fracture prediction has been studied extensively using both single parameter (K for brittle fracture and J for ductile fracture) and two-parameter (J-T and J-Q) characterizations. Both methods of fracture prediction are engineering approximations and thus have quantifiable limitations. A detailed understanding of the meaning behind the fracture parameters is necessary to determine the distinct limitations for their use. The linear elastic stress intensity factor (K) is only valid when the crack tip experiences only moderate plastic deformations. Once the crack tip deformations exceed the applicable limits of linear elastic fracture mechanics (LEFM), an elastic-plastic fracture parameter, the J-integral, is introduced to predict failure. Under geometric conditions in which the stress state is not determined by the single term K or J, a constraint parameter can be introduced to help accurately predict the stress state and therefore the fracture process. The use of a single asymptotic term is limited to geometries that maintain high constraint, such as the compact tension (C(T)) specimen where the inplane stresses are dominated by bending. The two-parameter approach also has its limitations. Many factors contribute to the validity range for two-parameter characterization. Geometric and loading conditions, material properties and bi-material interactions for structures with material discontinuities play important roles in the fracture process and in choosing viable fracture parameters. In this study, novel techniques are implemented to assess the geometric and material intricacies that a complex structure has on twoparameter characterizations. In particular, a liner/overwrap bi-material structure with a liner surface crack is investigated. Surface cracks are among the most common flaws 18

19 present in structural components and frequently the most critical in terms of limiting structural life, for both ductile and brittle materials (Leach et al., 2007; Evans and Riley, 1983). Methodologies for evaluating the effects of the elastic backing on the twoparameter characterization are developed using elastic-plastic finite element modeling. New techniques in constraint corrected stress field analysis are developed to evaluate hardening and stiffness material properties, loading conditions, and geometric constraints. Homogeneous uniaxial surface crack specimens are generated for the investigation thickness effects and comparison to more complex bonded models. The homogeneous uniaxial models are also used in a detailed study on the effects of plastic material properties, namely the strain hardening exponent in a power law hardening material, on two-parameter near tip field characterization. A preliminary study is implemented to investigate the effects of a structure containing a through thickness elastic-plastic to fully elastic material discontinuity in a uniaxial isotropic analysis. The elastic modulus ratio of the elastic-plastic cracked body and the fully elastic backing is also varied in this analysis giving a range of possible stiffnesses seen in a more complex geometry such as a COPV. The culmination of methodologies and material/geometrical investigations is the sub-scale and full-scale COPV models containing liner surface flaws and actual material properties. This study fully investigates two-parameter near tip stress field parameterization. Results from these studies will facilitate the implementation of geometric limits in testing standards and proof test logic for surface cracked specimens and COPV liners. 19

20 Composite Overwrapped Pressure Vessels The characterization of tip fields and fracture prediction of metal liners with surface cracks bonded to an elastically loaded structural backing has not been well established. The fracture prediction of such structures is of intrinsic and practical interest with wide application in bonded pressure vessel technology such as composite overwrapped pressure vessels (COPV). COPVs, used primarily in aerospace applications, generally consist of a thin (0.76+ mm) metallic (typically aluminum, titanium or stainless steel) inner liner wrapped with a comparatively thick (greater than three times the liner thickness) composite shell, commonly made from carbon/epoxy. The shell/wrap serves as the primary structural component, with the liner supporting very little load, but acts as a barrier for leakage of propellants and other pressurants. During the manufacturing process, COPVs undergo an auto-frettage pressure cycle in which the liner experiences plastic deformation in order to compensate for thermal strains induced from curing and the mismatch in thermal expansion between the liner and overwrap resulting in residual compressive stresses in the liner. An additional benefit of auto frettage is that the plastic strains accumulated in the liner will effectively extend the elastic range under operating conditions. Figure 1-2 gives an example of the stresses and strains during an auto-frettage cycle for an isotropic hardening material. The compressive loading comes from the overwrap elastically returning to its original shape, deforming very little from the stresses in the liner. The result is an operating range that can handle greater elastic strains. Under some extreme situations, the liner may undergo post manufacturing proof/operating plastic strain, but these loads will typically be for very low cycle operation. 20

21 Understanding the behavior of surface cracks in COPV liners is challenging because, (i) the liner may experience initial plastic deformation during manufacturing processes, (ii) the liner may experience cyclic plastic strains during normal operation, and (iii) surface crack behavior may be strongly influenced by the constraint effects resulting from the bonding. From a fundamental fracture mechanics perspective, the objective is to understand the constraint effects that arise in a thin liner bonded to a stiff composite and its role in characterizing crack tip fields. NDE, Proof Test Logic and Smart Cracks Non-destructive evaluation (NDE) is performed on all fracture-critical hardware. NDE methods are used to detect existing flaws. The existence of a detectable flaw deems the hardware unusable. The NASA standard NASA-STD-5009 (NASA, 2008) requires that all fracture-critical parts shall be subjected to NDE and/or proof testing to screen for internal and external cracks. NDE methods are selected for all part or component life cycles, including but not limited to manufacturing, maintenance, and operations. NDE detection techniques for cracks in metals include eddy current, penetrant, magnetic particle, radiographic, and ultrasonic. The minimum detectable crack size from these methods is assumed present in the structure at the most critical location. Simulated service life is then evaluated for fracture and fatigue. This method must demonstrate that the largest flaw not detectable by NDE can survive at least four mission lives. Proof testing and, if necessary, post-proof NDE are implemented to test hardware for service life given a verifiable proof safety factor from the above analysis. Proof testing begins with crack specimen testing and single parameter critical values or two-parameter failure locus. The limit of applicability and level of conservativeness can be assessed by experimental repeatability of test specimens, 21

22 ensured using ASTM standards, and a detailed numerical and analytical analysis of specimen and structure near tip stress field characterizations. Aerospace pressurized structures do not lend themselves to simplified techniques, such as LEFM, due to load levels up to net section yield. Additionally, aluminum and its alloys often exhibit the ductile tearing mechanism during crack growth. This mechanism yields large plastic zones and crack tip deformation making numerical and experimental predictions difficult. Therefore, proof test logic must be supported by a rigorous simulated service test program using detailed numerical simulations and experimental coupons to evaluate proof factors, failure loci and parameterization limits. Smart Crack refers to a flaw that passes proof/nde evaluations but progresses to unacceptable levels during normal operation. The smart flaw falls below critical limits achieved during proof testing yet extends and fails during normal operation. Complex geometric constraints resulting in highly non-linear crack response with growth, such as a surface flaw in the liner of a bonded geometry, can yield the possibility of a smart crack. Investigation into the meaning behind crack parameters, limit of their use and near tip field analysis can aid in preventing the existence of smart crack. Coupon and Subscale Testing Current testing for NASA s composite overwrap pressure vessel initiative includes the cyclic loadings of uniaxial surface cracked plate specimens and sub-scale COPVs containing liner surface flaws. Figure 1-3 is an example of one of the COPVs tested at the NASA Langley Research Center (LARC). Figure 1-4 shows one of many crack surfaces from a sub-scale COPV tested at NASA. The material is 6061-T6 aluminum. In addition, crack surfaces from many homogeneous uniaxial surface cracked coupons are available for analysis. 22

23 Figure 1-5 shows data from a COPV test specimen loaded under pneumatic pressure loadings. The strain measurements are taken from the center of the cylinder shown in Figure 1-3. The FEA results from a 3D elastic-plastic FEA model with anisotropic overwrap are shown and match well with the experiment. Fracture test specimens validate the more complex composite pressure vessel modeling implemented in a subsequent section. Using the global strain results and the fracture surface measurements, a numerical model can be constructed with confidence to verify the fracture parameterization of the liner crack. Further COPV sub-scale testing will be discussed in the final analysis section, Chapter 6. Two-Parameter Fracture Mechanics Plastic deformation precedes fracture in metals and their alloys and resistance to fracture is therefore directly related to the development of the plastic zone at the crack tip. Among the major causes of metallic structural failure is the nucleation and propagation of cracks from regions of high stress concentration such as notches and surface flaws, due to both monotonic and fatigue loading. Understanding the evolution of plasticity in notches and cracks is therefore important for predicting fracture behavior of critical load bearing structures in many engineering applications. Consequently, development of crack tip elastic-plastic fields, as a function of load configuration, geometry, monotonic and cyclic strain hardening behavior, and constraint effects have been subject of intensive study. Behavior of the crack tip in the plastic zone for strain hardening materials under small-scale yielding for symmetric (Mode I) or antisymmetric (Mode II) two-dimensional (2D) stress distribution has been presented in the widely referenced HRR articles by Hutchinson (1968a, 1968b) and Rice and Rosengren (1968). Extensions to 23

24 combinations of Mode I and Mode II loadings were presented by Shih (1973, 1974). Comprehensive literature reviews of plane-strain fracture mechanics and crack tip field characterization are found in Anderson (2004), Irvin and Paris (1971), McClintock (1971), Panayotounakos and Markakis (1991), and Rice and Rosengren (1968). The aforementioned works refer to fracture prediction using single parameter (K for brittle fracture and J for ductile fracture) crack tip field characterizations. The J- integral alone does not uniquely and accurately describe the crack tip fields and resistance against initiation of ductile crack growth when constraint effects arising from geometry and load configuration are considered. The constraint can be thought of as a structural obstacle against plastic deformation induced mainly by the geometrical and physical boundary conditions (Yuan and Brocks, 1991). Constraint effects can also arise from mismatch of material properties in a heterogeneous joint (English et al., 2010). Under these conditions, a second parameter, in addition to the J or K, is introduced to quantify the crack tip constraint. Crack tip triaxiality, which is defined as the ratio of hydrostatic pressure to von Mises stress, or a parameter that maintains a linear relationship with triaxiality, can be used as a constraint parameter for predicting ductile crack growth. The in-plane constraint is influenced by the specimen dimension in the direction of crack growth or the length of the uncracked ligament, and by global load configuration (bending or tension), whereas the out-of-plane constraint is controlled by the specimen dimension parallel to the crack front or the specimen thickness, for thorough-thickness cracks (Giner et al., 2010). Out-of-plane constraint is typically denoted as a plane-strain (high triaxiality) or plane-stress (low triaxiality) state. In finite thickness fracture geometries, where the points are embedded entirely in material, there 24

25 exists a state of plane-strain near the crack tip, the exception being very thin films (O Dowd, 1995; Shih et al., 1993; Wang, 2009). J-T and J-Q Characterization of Near Tip Fields Williams (1957) showed the existence of a non-singular in-plane normal stress component (T-stress) for linear elastic material. The significance of the T-stress on the size and shape of the plastic zone under small-scale yielding (SSY) conditions was shown by Larson and Carlsson (1973) and Rice and Tracey (1974). O Dowd and Shih (1991, 1992) and Sharma and Aravas (1991) show the important role played by higher order terms in asymptotic solutions of crack tip fields and demonstrate that a twoparameter characterization of the crack tip fields involving J and a triaxiality or constraint parameter Q is necessary to satisfactorily describe the configuration dependence of fracture response of isotropic plastic solids, particularly under large scale yielding conditions. The use of a single parameter characterization (K or J) is limited to geometries that maintain a high constraint, such as the compact tension (C(T)) specimen, where the in-plane stresses are dominated by bending. The Q stress difference factor has been found to maintain a material dependent linear relationship with near tip triaxiality independent of geometry, dimensions and deformation level. Moreover, the Q factor is shown to accurately predict near tip stress and strain fields, particularly plastic strains, and therefore can be used as a ductile fracture parameter (Henry and Luxmoore, 1997; Kikuchi, 1995). A significant body of work has focused on two-parameter characterization of through-thickness cracks while limited work has been done on surface crack geometries. Wang (1993) established the J-T characterization with modified boundary layer (MBL) reference solutions to topographical planes perpendicular to the crack front 25

26 in surface crack tension (SC(T)) and surface crack bend (SC(B)) geometries. Wang (2009) investigated Q as a function of load and radial distance from the crack tip for surface crack uniaxial and biaxial tension models. A J-T or J-Q family of fields generated with MBL formulation is used as a comparison for assessing near tip dominance in actual structures. A near tip region is said to have J-T or J-Q dominance if the stress and strain distributions match the MBL reference field when the length scale is normalized by J/ζ o, where ζ o is the yield stress. Shih et al. (1993) and O Dowd (1995), in addressing the limitations of plane-strain references applied to three-dimensional geometries, suggested that as the distance from the crack tip becomes small, the out of plane (parallel to the crack front) strains become negligible when compared to the in-plane singular fields. Therefore, planestrain MBL prediction of surface crack front stress fields should be accurate as the distance from the crack tip approaches zero. Fracture Prediction Two-parameter fracture mechanics is important in engineering applications because it provides a more adjusted assessment of failure limits (Cicero et al., 2010; Silva et al., 2006). The primary source of enhancement stems from the constraint effects on near tip plastic deformations and subsequent effect on critical loads. Iwamoto and Tsuta (2002) and Li and Chandra (2003), in their studies of crack growth characteristics in ductile materials, describe the direct influence of plastic strain and strain-induced material transformations at the crack tip on apparent fracture toughness. Faleskog (1995) showed experimentally how an increase in the constraint factor Q results in a decrease in the critical value of J. Chao and Zhu (2000) and MacLennan and Hancock (1995) numerically and experimentally verified this effect on J-resistance 26

27 (J-R) curves and failure assessment diagrams (FAD) respectively, with the highest constraint geometries having the lowest resistance. Leach et al. (2007), in a study using J and an average opening stress constraint parameter to predict critical crack growth angles in surface crack tension SC(T) and bend SC(B) specimens, verified the enhanced accuracy of a constraint modified fracture criterion. Similar results can be found throughout literature and it is now generally accepted that the increase in energy dissipation through plastic deformation due to loss of constraint can account for the increase in apparent fracture toughness. Deformation Limits Many investigators have examined single and two-parameter characterizations limits (Kim et al., 2003; Larsson and Carlsson, 1973; McMeeking and Parks, 1979; O Dowd and Shih, 1991, 1992; O Dowd, 1995; Varias and Shih, 1993; Wang, 1993; Wang and Parks, 1995; Wang, 2009; Zhu and Chao, 2000). In single parameter fracture mechanics, the dominance limit of the asymptotic term is controlled by geometry and load/deformation. With the introduction of a non-singular geometry dependent constraint term, such as T or Q, the limits of characterization can be expressed by a load or deformation term. Wang and Parks (1995) considered the parametric limits of the constraint corrected asymptotically singular near tip stress fields based on crack tip deformations measured by J/ζ o relative to a characteristic length l, such as the remaining ligament or crack depth. The limiting relative local deformation factor lζ o /J cr, where J cr is the J-integral at which two-parameter dominance of near tip fields cannot be assured, has been suggested by many investigators and is a requirement in ASTM E1820 for compact tension (C(T)) and single edge notched bend (SEN(B)) specimens (ASTM, 2006). This factor is dependent on loading conditions and 27

28 material parameters, but given a proper definition of l and methodology to determine field dominance, a conservative limit locus can be found. Geometries with deformation limit loads below critical fracture load will often have inflated toughness values with increased scatter. However, stress fields predicted using a micromechanical model within two-parameter deformation limits can be used to construct a constraint modified locus for cleavage fracture (Faleskog, 1995). In addition to experimentally establishing J and Q as accurate predictors of cleavage fracture in surface crack tensile (SC(T)) specimens, Faleskog (1995) provides a reasonable verification for the J-Q approach. He observes that the hoop stress (ζ θθ ) deviator of the Q defined difference fields is relatively consistent along the crack front for loads causing cleavage fracture, providing verification that Q is a proper measure of triaxiality. In addition, he uses MBL predicted opening stress fields to prove the existence of J-Q dominance in the crack plane at the cleavage fracture initiation positions along the crack front. The preliminary limitations analysis by MacLennan and Hancock (1995) of opening stress fields compared to MBL predictions is used to validate their experimentally constructed modified FAD approach. To achieve a proper verification of two-parameter deformation limits, micro-scale deformation data and detailed near tip stresses are necessary. Therefore, finite element analysis is used to extrapolate experimental data in order to verify the existence of J-Q fields, as in MacLennan and Hancock (1995). Furthermore, the loss of two-parameter near tip field dominance can only be predicted using numerical methods and can be observed secondarily from various experimental results. 28

29 Paul and Khan (1998) in studying finite element models of a centrally cracked thin circular disk show that the inability to maintain hydrostatic pressure accounts for the reduced stress state at the crack front even with small strain approximations, but uncontained hydrostatic pressure very close to the crack tip results in high triaxiality and a maximum opening stress hump. This region, denoted as the process zone, is where damage from void coalescence and crack growth commonly takes place (Xia and Shih, 1995). This phenomenon can help to explain the near tip stress field variances from MBL predicted stresses, namely the rapid opening stress reduction at distances after the hump for large levels of deformation. The importance of including effects of hydrostatic stress and the third invariant of the stress deviator on stress state computation has been shown by Bai and Wierzbicki (2008) and Gao et al. (2009, 2010) since it can play an important role on both the plastic response and ductile fracture behavior of certain aluminum alloys. Ductile fracture analysis can be used subsequent to or in conjunction with traditional fracture parameterizations and has been the topic of much interest (Li et al., 2010; Mirone and Corallo, 2010; Wei and Xu, 2005; Xue et al., 2010). Li et al. (2010) give a thorough analysis of ductile fracture computational techniques and experimental results for a power-law material in which the undamaged response follows J 2 plasticity theory. Although this initial isotropic material model is widely used for numerical evaluation of fracture parameters and near tip stress field analysis, ductile fracture techniques using void growth, shear band and anisotropy are necessary for accurate prediction of critical failure beyond limit deformations (Huespe et al., 2009; Sun et al., 2009). These types of analyses are important for the failure assessment of structures 29

30 undergoing plastic deformations during normal operation in which the combination of geometry and plastic material properties prevents single and two-parameter fracture prediction but add a greater level of complexity. 30

31 Figure 1-1. Examples of common COPV geometries Auto-Frettage 0.5 / o Operating Range / o Figure 1-2. Stress-strain curve for the auto-frettage cycle of a COPV 31

32 Figure 1-3. Picture of a COPV used for fracture testing 2c f = 1.12 mm 2c o = 1.02 mm a f = mm a o = mm Figure 1-4. Surface crack face from a subscale COPV liner t = 2.29 mm (0.09 in.) 32

33 Hoop FEA Longitudinal FEA Hoop Experiment Longitudinal Experiment Pressure (MPa) Figure 1-5. Outer strain gage measurements from COPV test specimen with FEA comparisons 33

34 CHAPTER 2 BACKGROUND Linear Elastic Fracture Mechanics In modern fracture mechanics, crack sizes determined by the resolution of nondestructive evaluation techniques are often assumed to be in a structure. The stability of an existing crack is the primary focus of fracture mechanics. For sharp cracks in an isotropic linear elastic material, the stress state approaching the crack tip produces a singularity. Linear elastic fracture mechanics (LEFM) is the practice of design in such a situation. Understanding the basics of LEFM and its limitations is necessary to assess the fracture problem of more complex materials and structures. The loading of a sharp crack in an isotropic linear elastic material produces a 1 r singularity. This stress state is proportional to the stress intensity factors K I, K II and K III for mode I, II and III (opening, sliding and tearing modes) respectively. The stress state is defined by ( I ) ij ( ( II ) ij III ) ij K r ij (2-1) 2r I ( I ), f K r ij (2-2) 2r II ( II ), f K r ij (2-3) 2r III ( III ), f where (I ) f ij, (II ) f ij and (III ) f ij are dimensionless functions of θ that are representative of the angular variation in stress for mode I, II and III loadings respectively. Williams (1957) showed that the above stress state is a series, where the first two terms under mode I loading are 34

35 ij K (2-4) 2r I r, fij T1 i1 j where T, called T-stress, is an elastically scaled parameter that plays an important role in elastic-plastic fracture mechanics and will be discussed in the subsequent section. The linearly scalable stress intensity factor can be calculated in closed-form for simple configurations. However, numerically determined fits and experimental methods are needed for geometries that are more complicated. The use of a single parameter to fully describe the stress state suggests that a material dependent critical stress intensity factor exists that will define the failure state of any crack in a structure. This practice has been used for many decades and is verified by extensive testing and standards, but has quantifiable limitations. In a real material, the infinite stress prediction at the crack tip will not exist. The result is crack tip blunting due to plastic deformation. For moderate levels of plastic deformation, corrections are needed to use LEFM, but the K-predicted stress state is still valid for regions sufficiently far from the plastic zone. Therefore, under certain conditions the stress intensity factor K can be used as a single term to predict failure. The size of the plastic zone compared to relevant structural dimensions must be small. Past emphasis has been placed on plane-strain fracture for test specimen geometries. It is commonly accepted that the lower bound critical fracture toughness is found under plane-strain conditions. The result is a plane-strain fracture toughness (K Ic ), which is considered a unique material property. The actual state of lower bound critical fracture is more closely related to the triaxiality or constraint at the crack tip, where the apparent fracture toughness is greater for lower constraint geometries and loadings. Since the state of plane-strain has high crack tip triaxiality the resulting 35

36 toughness values would appear smaller. Constraint will be discussed in a subsequent section. Elastic-Plastic Fracture Mechanics For high toughness, low strength materials (high K c and low ζ y ) the criterion for LEFM are difficult to satisfy. Large plastic zones before fracture invalidate LEFM assumptions. A different approach to fracture mechanics is necessary to account for the larger plastic deformations at the crack tip. Elastic-plastic fracture mechanics assumes crack growth occurs under stable ductile tearing and terminates by tearing instability. A new term that does not rely on the 1 r crack-tip singularity and accounts for the plastic deformation is necessary to predict fracture. The J-integral is defined as a path independent contour integral about an arbitrary path Г enclosing the crack tip: J ui wdy Ti ds (2-5) x where: w = strain energy density T i = components of the traction vector u i = displacement vector components ds = length increment along the contour Г w ij 0 ijd ij (2-6) T i n (2-7) ij j where n j are the components of the unit vector normal to Г. The units for J are energy per unit area; therefore, it is defined in an elastic material as the energy release rate (the energy released for a unit area of crack growth). Much like the stress intensity 36

37 factor K, the J-integral can be used as a single parameter for characterizing crack front fields. The benefit of a J-integral approach verses other elastic-plastic methods (such as crack tip opening displacement or CTOD) is that under small-scale yielding (SSY), the J-integral simplifies correlatively to K. A critical value of J exists to characterize fracture if the asymptotic fields about the crack tip are characterized by this single parameter. This only happens when constraint is zero and SSY conditions apply. As large-scale plasticity becomes prevalent, the stress field begins to deviate from the SSY curve. For a given crack-tip with plasticity, annuli measuring the extent of asymptotic dominance can be expressed. The annulus near the tip within the area affected by crack tip blunting is represented by plastically dominated large strains, then extending radially the J-dominated region and finally the K-dominated elastic field under SSY. Dominance refers to the state in which the fracture parameters accurately represent the near crack stress state for a given geometry and loading condition. Two-Parameter Characterization Under large-scale yielding, geometry and size considerations in fracture become necessary. Much research has been done on the geometric effects on near crack stress fields (Leach et al., 2007; Wang and Parks, 1995; O Dowd and Shih, 1991, 1992; Sharma and Aravas, 1991; Rice and Rosengren, 1968; Betegón and Hancock, 1991; Wang 1993; Wang and Parks, 1995; Newman et al., 1993; Shih et al., 1993; O Dowd, 1995). Two-parameter characterization methods exist to help predict failure by addressing the geometrically determined constraint of a structure. Figure 2-1 demonstrates the relationship between constraint, level of deformation and fracture parameterization necessary to characterize failure. Constraint is denoted 37

38 by ξ and can be either T or Q (ξ o corresponds to T = Q = 0). The loading curves and critical points are shown for various testing standards. Examples from the Linear Elastic Fracture Toughness (E399), Elastic-Plastic Fracture Toughness (E1820) and the Surface Crack (E740) standards are shown. For a constraint sensitive material, the measured J c will be greater for low constraint test specimens such as middle crack tension (M(T)) or single edge notched tension (SEN(T)). Lower bound fracture toughness is often found using a high constraint geometry such as the compact tension (C(T)) specimen. For conservative lower bound fracture, the C(T) specimen is often used, but for a more adjusted design, a constraint corrected failure criteria is needed. Many investigators have proposed the existence of a single failure measurement that is a function of both asymptotic and constraint parameters to predict crack growth and fracture (Leach et al., 2007; Wang and Parks, 1995). For actual geometries, fracture predictions using standard test specimens are reliant upon the ability to predict the stress field within the damage zone around the crack tip. The limit of stress field prediction can be expressed as a region (annulus) around the crack tip or a load for a specific geometry and loading condition. Within this limit, stress fields can be compared directly, and damage prediction can be made from single (high constraint) and two-parameter (low-constraint) measurements. O Dowd and Shih (1991, 1992) and Sharma and Aravas (1991) demonstrate that a two-parameter characterization of the crack tip fields involving J and a constraint parameter is necessary to describe the stress fields under large-scale yielding conditions. Rice and Rosengren (1968) also notes the strong configuration dependence 38

39 of the near-tip deformation fields, and that the asymptotic solution may not be applicable under large-scale yielding conditions to low-constraint fracture geometries such as center crack tensile (CC(T)) or single edge notch tensile (SEN(T)) specimens, where the stresses around the crack tip may be much lower. Two-parameter characterization uses a measure of the asymptotic fields, such as the stress intensity factor K, crack-tip opening displacement (CTOD) or the J-integral with an additional constraint parameter, such as the T-stress or stress difference Q, to predict crack-front stress fields. For loads causing deformation beyond the dominance limit, the near crack stress fields are considered plastically collapsed. Determining twoparameter dominance relies on the ability to model idealized stress fields within the characterized range. This is done by use of modified boundary layer (MBL) solutions. MBL models are used to create a reference space to compare the stress fields with an elastic-plastic material. J-T Theory The Williams expansion of a linear elastic stress field is a series with the leading term having a 1 r singularity and the second term being constant with r. The second term is defined for mode I crack opening as the T-stress. Equation 2-4 shows the first two terms of the Williams expansion for mode I loading. T-stress is defined as a uniform stress parallel to the crack plane and perpendicular to the crack front. Low crack-tip constraint is characterized by negative T-stresses. Geometries with negative T-stresses do not maintain single parameter J-dominance because constraint loss is associated with a lowering of the stress state. However, negative T-stress geometries can be described by the J-integral and T-stress up to and exceeding net section yield 39

40 (Betegón and Hancock, 1991). T-stress is not defined under fully yielded conditions; nevertheless, elastically scaled T-stresses calculated from far-field loads in elasticplastic models are a relatively accurate constraint measurement for two-parameter characterization (Wang, 1993). T-stress is a measure of crack tip constraint (triaxiality); therefore, it changes the crack tip stress field conditions such as the opening stress (stress perpendicular to the crack plane) distributions and plastic zone shape and size. O Dowd (1995) showed the strong constraint dependency on plastic strains near the crack tip for various power-law hardening materials. Constraint or triaxiality directly influences the size and shape of the plastic zone. Figure 2-2 demonstrates how normalized T-stress (T/ζ o ) changes the plastic zone shape. Low T (T/ζ o = -1) results in a tendency away from the crack face while high T (T/ζ o = +1) shifts the zone towards the opening face. Zero T/ζ o results in a plastic zone common of the plane-strain SSY condition. The stress ζ o is defined as the reference yield stress; this is typically taken as the 0.2% offset yield (ζ o = ζ y ) unless otherwise noted. The applicable limits of J-T two-parameter characterization of elastic-plastic cracktip fields using modified boundary layer (MBL) solutions have been explored by many investigators (Newman et al., 1993; Wang and Parks, 1995) for through-crack specimens. Wang (1993) extended the two-parameter characterization with MBL reference solutions to topographical planes perpendicular to the crack front in surface crack tension (SC(T)) and surface crack bend (SC(B)) geometries. In addition to verifying the applicability of MBL solutions for J-T limit determination in surface cracked plates, this study quantitatively explains the short falls of plane-strain characterization. 40

41 The general conclusion was that a completely consistent two-parameter description of crack-tip stress fields is not possible beyond certain loads but is effective in determining the deformation limits. Elastically scaled T-stress factors are readily available in literature for a number of geometries making it a valuable measure of constraint for engineering applications. However, in large scale yielding conditions, this measure of constraint loses its physical meaning. J-Q Theory The easily applied elastically scaled T-stress factors are useful in engineering applications; however, a direct measure of the stress field within the plastic zone (Q) is a more accurate measure of constraint for geometries undergoing large-scale plasticity. J-Q theory has been implemented for the characterization of near tip stress fields for 2D and 3D geometries by many investigators. Wang (2009) shows that the Q parameter as a measure of the difference between the actual opening stress and the small-scale yielding solution is accurate for a range of normalized radial distances (r/(j/ζ o )) for loads exceeding yield, where the ratio J/ζ o scales with the blunting zone diameter or crack tip opening displacement. The constraint parameter Q is defined as the difference between the actual hoop stress (ζ θθ ) and a reference hoop stress within a cracked body. The hoop stress (ζ θθ ) is defined as the normal stress in the direction tangent to a circular region enclosing the crack tip. Figure 2-3 gives a definition of the θ direction for the MBL models. O Dowd and Shih (1991, 1992) showed that the following equation can be used within a range of normalized radial distances, typically 1.5 r/(j/ζ o ) 5 to accurately represent the near crack fields: 41

42 ij r, J for -π/2 θ π/2 (2-8) o r ij, Q oij T 0 where the first term is the SSY solution and δ ij is the Kronecker delta. The SSY solution is used over the HRR solution, first proposed by O Dowd and Shih (1991, 1992), to reduce the radial distance dependency (O Dowd and Shih, 1994). In the analyses presented by Silva et al. (2006), plane-strain results produce relatively unchanged Q values when the normalized radial distance from the crack tip is varied between 1 r/(j/ζ o ) 5. Similarly, in three-dimensional crack front fields of surface crack specimens, where the points are embedded entirely in material, there exists a state of plane-strain near the crack tip. Therefore, variations of the normalized radial distance within this range for J-dominant surface crack fields should not change the general trends presented in the analysis or the conclusions derived from them. However, Bass et al. (1999) shows that under biaxial loading, the opening stress difference fields do not correspond to the hydrostatic shift proposed by O Dowd and Shih (1991, 1992) indicated by the dependencies of Q on radial distance. Similarly, Wang (2009) shows that for deep surface cracks (a/t = 0.6) under biaxial and uniaxial tension, Q fails to remain radially independent for angles close to the free surface at large deformations. These considerations make a thorough analysis of the applicable limits of Q as a ductile fracture parameter in pressure vessel geometries vital to predict crack growth and failure. The Q-parameter is best defined as a fracture parameter when the normalized radial distance equals 2J/ζ 0 because this marks the location where cleavage mechanism is triggered ahead of the crack tip (Silva et al., 2006). Therefore, in order to 42

43 be used in two-parameter fracture characterization, Q is defined as the hoop stress difference in the cracked plane (opening stress or ζ yy ) at a radial distance equal to 2J/ζ o : Q o yy yy T 0 at θ = 0 and r = 2J/ζ 0 (2-9) A single value of Q represents a material dependent near crack stress field as a function of r/(j/ζ o ). Using this definition of constraint, a plane-strain J-Q family of fields is constructed using the MBL finite element formulation. Nevertheless, the relative changes in Q with radial distance (r) when measured at larger deformations, necessitates the investigation of opening stress as a function of normalized radial distance r/(j/ζ o ). The deformation limit of J-Q dominance may therefore be determined by the relative change of opening stress compared to those predicted at r = 2J/ζ 0 or when the radial independence of Q cannot be assured (Sharma et al., 1995). Modified Boundary Layer Formulation Determining two-parameter dominance of a test specimen or structure relies on the ability to model idealized stress fields within the characterized range. Where analytical solutions exist to compare stress states in fully elastic cracked bodies, the elastic-plastic fields defined for an arbitrary material must be formulated using MBL finite element models. The MBL stress field solutions are derived from the existence of an asymptotic plane-strain elastic stress field outside crack-tip plastic zone under contained plasticity. This field is uniquely defined by the elastic parameters (K and T). Contained plasticity is defined as the state in which the plastic zone is on a length scale that is small compared to relevant dimensions. Under these conditions, the J-integral simplifies correlatively to 43

44 K, and therefore a region exists within the plastic zone that must be defined by the elastic-plastic parameter J and a measure of constraint (T or Q). Moreover, the unique fields described by this state are representative of the stress state in any crack geometry and load configuration that is defined by these parameters. SSY conditions are a special case in the MBL formulation where the stress field measured outside the plastic zone but far from the geometry boundaries is characterized by the first singular term of the Williams Eigen expansion (Williams, 1957; Wang, 1993): ij K, ij (2-10) 2r I r f where K I is the mode I stress intensity factor. A material flow curve (ζ verses ε), typically defined by uniaxial tension data, and the Poisson ratio are the only variables needed to produce a unique set of MBL reference fields. The MBL solution is therefore a derived material property. An accurate MBL reference field should be reproducible when it is independent of K- applied and mesh configuration as long as the formulation follows certain criteria defined later in this section. The remote tractions for MBL formulation are given by the first two terms of the Williams Eigen expansion (Williams, 1957): ij K (2-11) 2r I r, fij T1 i1 j where K I is the mode I stress intensity factor and T is the T-stress. Figure 2-3 shows the half-symmetric plane-strain FE mesh used for MBL analysis. The first two terms of the Williams series are applied as displacement boundary conditions to the r = r max outer surface. The corresponding in-plane displacements are given by: 44

45 u i K r T i (2-12) E 2 E I r g, rh,, i where E and ν are material constants and g i and h i are the angular variations in displacement caused by elastic singularity fields and T-stresses respectively (Wang and Parks, 1995). The angular variations are defined in plane-strain as: g x, 1 cos 3 4 cos (2-13) 2 g y, 1 sin 3 4 cos (2-14) 2 x 2, 1 cos h (2-15) y, 1 sin h (2-16) Plane-strain MBL solutions for elastic-plastic crack tip fields can be obtained by applying these displacements to the outer boundary of the circular crack tip region and varying the T-stress while K I remains constant. The J-integral is related to the stress intensity factor K I in plane-strain by: J K I (2-17) E Abaqus version 6.7 is used to compute the MBL solutions (Dassault Systèmes, 2007b). Figure 2-4 shows the near crack opening stress fields predicted by the MBL model for a 6061-T6 aluminum. Wang (1993) showed that these MBL solutions provided accurate prediction of stress fields in surface crack tension (SC(T)) specimens as characterized by J and T, and that the crack tip fields of the MBL solution far from the outer boundary and outside the crack tip blunting zone should represent those of any crack with the same K I and T. 45

46 While an elastically scaled T-stress can be used to accurately predict near tip stress fields up to and exceeding net section yield, the physical meaning of T is lost with the onset of large-scale plastic deformation. Therefore, the Q factor, which is calculable at loads exceeding yield, is often used in stress field prediction. Figure 2-5 shows the unique relationship between T and Q in the MBL solutions. The near tip reference stress fields are often formulated as a function of constraint for constant normalized radial distances. Figures 2-6 (A) and (B) gives these curves for r/(j/ζ o ) = 2, 4, 6, and 8 as functions of T and Q respectively. Modified boundary layer solutions can only be formulated within certain constraint levels. As the compressive constraint stresses (negative T-stress) approach the yield stress, crack tip deformations exceed the contained plasticity limits and no longer provide an accurate representation of the J predicted field. Literature has shown the applied T-stress limits to be approximately T = -ζ o, but the actual limits can be determined by close examination of the near tip conditions. When the crack deformations exceed levels necessary for MBL formulation, the elastic-plastic J-integral (J) will deviate from the elastically predicted value (J El ), given by Equation 2-17, by a considerable amount. This can be formulated as an arbitrary limit where J must not deviate from J El by greater than 10%, as shown below: J El J J El 100% 10% (2-18) One must also consider the limitation of MBL formulation as a function of applied K. With increasing applied K and the onset of large-scale plasticity at a low level of constraint (T -ζ o ), the J-integral deviation, as defined by Equation 2-18, increases exponentially. Nevertheless, it is necessary to apply a sufficiently large K to achieve 46

47 radial data points close to the crack tip. Since K is normalized out for the final reference solutions, repeatability remains solely on adherence to the requirements of Equation 2-18, which also ensures deformations below large-scale plasticity. Therefore, the MBL solutions cover the desired level of constraint (0 T/ζ o -0.9) and a maximum K is applied such that the error of J at the minimum applied T-stress (T/ζ o = -0.9) is within the above limits. Near Tip Stress Field Analysis and Plastic Collapse The near tip stress fields are analyzed in the cracked plane along the vector normal to the crack front in the growth direction. The radial distance r from the crack front is normalized by J/ζ o which gives a length scale comparable with MBL solutions. The nodal opening stress values are used along with cubic spline interpolation to determine the near tip stress distribution. From this curve, the value of Q is determined with Equation 2-9. The unique relationship of Q to a specific near tip field provides the reference solution for comparison. Figures 2-7 (A) and (B) give examples of near tip opening stress fields verses normalized radial distance (r/(j/ζ o )) for high (n = 3) strain hardening and low (n = 20) strain hardening materials respectively as defined in a linear plus power-law material model. The J-Q MBL predicted stress field and the neutral constraint (SSY) stresses are given as references for the actual stress state in a surface crack model with a/c = 1.0 and a/t = 0.35 at crack depth (ϕ = 90 ) for a far field load ζ = ζ o. Two processes exist by which near tip stress field parameterization is lost. First, single parameter dominance is limited in low constraint geometries by a loss of constraint or triaxiality near the tip resulting in significant deviation from the SSY solution, even under contained plasticity. Secondly, two-parameter dominance is limited by the amount of 47

48 deformation at the crack tip and the subsequent shape change and/or rapid relaxation of the fields compared to those predicted by MBL solutions. This process is referred to as plastic field collapse. Because the stress fields in J-Q characterization are referenced to a point near the crack tip (r = 2J/ζ o ), field collapse is measured as difference at larger radial distances or slope variation from the predicted field. 48

49 Collapse Increasing Deformation J-ξ J-ξ or K-ξ E399 K Ic Example E1820 J Ic Example E740 Examples Low constraint requires 2 parameters to describe fields Large Scale Yielding Small Scale Yielding ξ o J J or K ξ High constraint requires 1 parameter to describe fields Figure 2-1. Diagram showing the relationship between constraint and deformation for various fracture characterizations T/σ o = 0 T/σ o = 1.0 T/σ o = -1.0 Figure 2-2. Plastic zone shapes diagram for high (T/ζ o = 1), zero (T/ζ o = 0) and low (T/ζ o = -1) T-stresses 49

50 u y r u x y θ x Figure 2-3. Modified boundary layer (MBL) model mesh and displacement boundary conditions T/ o = 0, SSY 2.5 yy / o T/ o = r/(j/ o ) Figure 2-4. MBL generated near tip opening stresses as a function of normalized radial distance r/(j/ζ o ) for a series of T-stresses ranging from T/ζ o = 0.4 to

51 Q T/ o Figure 2-5. Relationship between normalized T-stress and Q r/(j/ o ) = 2, 4, 6, r/(j/ o ) = 2, 4, 6, yy / o 2 yy / o A T/ o B Q Figure 2-6. Reference opening stress as a function of T/ζ o (A) and Q (B) for normalized radial distances r/(j/ζ o ) = 2, 4, 6, and 8. 51

52 yy / o SSY (MBL) A 3 SC(T) 2.5 / o = 1.0 a/c = a/t = 0.35 J-Q (MBL) = r/(j/ o ) SSY (MBL) 2.75 yy / o J-Q (MBL) B 2 / o = 1.0 SC(T) a/c = a/t = 0.35 = r/(j/ o ) Figure 2-7. Examples of the near tip opening stress fields at a constant far field stress ζ = ζ o as a function of normalized radial distance r/(j/ζ o ) for a semicircular surface crack (a/c = 1.0 and a/t = 0.35) with the zero constraint small-scale yielding (SSY) and the constraint corrected J-Q predicted fields shown as solid points for materials with n = 3 (A) and n = 20 (B). 52

53 CHAPTER 3 FINITE ELEMENT MODELING AND TEST GEOMETRIES Geometries Surface cracked plates under monotonic tension are the primary focus of this study. Three configurations are used to model this geometry. The first and the most general model tested is an uniaxial surface cracked specimen with far-field displacement (δ ) or stress (ζ ) boundary conditions acting normal to the crack face, shown in Figure 3-1. Similar to this configuration, the second model tested is the uniaxial surface cracked plate specimen bonded to an elastic plate on the face opposite crack opening (Chapter 4). Lastly, a surface cracked sub-model section is implemented from a global metallic pressure vessel or COPV geometry. The crack is oriented such that it experiences mode I opening in the direction of maximum normal stress in the liner (hoop in the cylindrical portion), considered to be the most critical in terms of ductile crack growth. The combination of specimen geometries and complex biaxial structures facilitates a complete understanding of the various geometric and material effects. The finite element crack geometries contain two planes of symmetry. The first is the xz-plane at the crack in which the out of plane displacements in the remaining section are set to zero (u yy = 0) and the second is the central yz-plane bisecting the crack mouth in which u xx = 0. A more detailed description of geometries and boundary conditions will be given in the following chapters. Mesh Configuration Finite Element Models FEA Crack version (Quest Reliability, 2007) was used to generate the crack mesh and Abaqus version 6.7 (2007b) was used for the analysis. Both an elastic- 53

54 plastic model and a fully elastic model for each of the geometries tested are necessary when using T as the constraint parameter. The elastic-plastic model is used to calculate opening-stress fields and J-integrals. The fully elastic model is used to find the T-stress scaling factors (T (ϕ)), which are normalized by the nominal stress in the opening direction of the metallic section. The J-integrals along the crack front are calculated using the domain integral method applied to three dimensions intrinsic to Abaqus. The T-stresses are calculated using the interaction integral method intrinsic to Abaqus in the fully elastic models (Dassault Systèmes, 2007a). The finite element analyses are made using C3D20R 20-node isoparametric brick elements. At the crack-tip, 20-node collapsed face prismatic elements are used. To allow for crack tip blunting in the elastic-plastic models, the initially coincident nodes at the crack tip are left unconstrained. Figure 3-2 schematically illustrates the collapsed face element nodal configuration and boundary conditions for elements with a face in the crack plane in the crack growth direction (θ = 0 ). In order to maintain sufficient data points at the desired radial distances, the crack region contains a highly refined mesh consisting of 31 contours around the crack tip and 72 collapsed face elements along the crack front. Figure 3-3 shows the crack face mesh with these refinements. Mesh Convergence A preliminary study of mesh refinement was conducted by comparing domain integral (J-integral) values, which tend to be sensitive to mesh refinement, from various models of mesh density. With increasing mesh refinement, the mean J-integrals at each crack angle (J(ϕ)), calculated by averaging the values from the outer most contours in which the J-integral varied by a negligible amount (< 0.10 %), converged to stable values and continued through the mesh refinement used for this analysis. The 54

55 chosen mesh refinement is beyond the level needed for accurate stress and domain integral calculations, but was necessary to achieve sufficient radial data points to assess the opening stress field using interpolation methods. Increasing the number of nodes in the radial direction from the crack-tip will decrease the maximum possible error in interpolation within numerical limitations. The near tip stress measurements obtained by a less refined mesh compared accurately to the results from the final mesh refinement (Figure 3-3) when the number of radial points in the coarse mesh used in interpolation is maximized by limiting the range of deformations observed. In other words, an accurate comparison can be made between meshes when the radius of near tip mesh refinement is approximately equal to 10J/ζ o, resulting in the maximum number of stress measurements. Outside this range, the coarse mesh does not contain sufficient data points for a meaningful comparison. Element Formulation Crack tip blunting effects can typically be neglected at distances greater than roughly one and a half to twice the crack-tip opening displacement (approximately 1.5J/ζ o or 2J/ζ o ). Inside this distance from the crack tip, it would be necessary to use large strain analysis (Betegón and Hancock, 1991; Wang, 2009). Linear kinematic element formulation (small-strain) simplifies the analysis because it does not require a finite radius crack-tip and often converges with a relatively coarse mesh. Non-linear kinematic element formulation (large-strain) requires a finite radius crack-tip and therefore mesh convergence is often difficult to achieve. For this investigation, smallstrain approximation is used to analyze stress fields outside the crack-tip blunting zone in the surface cracked models. Figure 3-4 (A) gives an example of how the small strain approximation of opening stresses differ from those calculated using large-strain 55

56 analysis with a finite radius key-hole mesh configuration at the crack-tip. This figure gives opening stress data from a single edge notched tension (SEN(T)) model (crack depth to width ratio: a/w = 0.20) with plane-strain boundary conditions at a load aζ o /J = 40 using the two linear plus power-law material models with n = 3 and 20. Figure 3-4 (B) gives the percent difference between the large and small-strain results. Percent difference is defined as Large Strain SmallStrain yy yy % Difference 100% Large Strain yy (3-1) Both linear plus power-law low and high strain hardening materials, n = 20 and 3 respectively, achieve 95% opening stress accuracy within the radial distance r = 2J/ζ o. Additionally, the higher strain hardening material achieves 95% accuracy at a smaller radial distance (r = 1.5J/ζ o ) compared to the model with the low strain hardening material. 56

57 Figure 3-1. General uniaxial surface crack model configuration. The applied loads, ζ and δ, are the average far-field (nominal) stresses and displacements respectively in the y direction. 57

58 y θ x Free Nodes (Crack Face) Symmetry BC (u yy = 0) 2D view of crack front element configuration ,7,4 6,8 1,5,2 Pre-collapsed face configuration Undeformed, collapsed face configuration for crack front elements Figure 3-2. Example of crack front collapsed face nodal configuration. For elements with a face in the θ = 0 plane, nodes 2, 3, 5, 6, and 7 are free while displacement u yy = 0 for nodes 1, 4 and 8. All other initially coincident nodes are free for remaining elements in the θ direction. 58

59 Figure 3-3. Finite element surface cracked mesh refinement detail. 59